Finite Group Databasebeta

← #13160095

Subgroups — #13160095

Order 1728.

1112 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89018C211yesmaximal, characteristic
2C2×C32×Dic12C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592886—31nomaximal
9#10590966—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6292—21no—
15C2×#6289—21no—
16#89018—22no—
17#89018—22no—
18C2×C33×C8C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×Dic12—22no—
22#88806—21no—
23#88803—21no—
24#88795—22no—
25C2×#3832—31no—
26C2×#3799—34no—
27C2×C32×Q16—31no—
28#64155—31no—
29C2×#3755—31no—
30C2×C3×Dic12D314yescomplemented
31#63856—34no—
32C2×#3717—34no—
33#63662—31no—
34#63648—31no—
35#63648—34no—
36#63648—64no—
37#63647D311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39#63566—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))D412yes—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6292—24no—
56#6289—24no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5187—94no—
60#4733—91no—
61C2×C32×Q8—31no—
62C2×C32×Q8—31no—
63#3832—62no—
64#3832—62no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—34no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—64no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×C3×C4×D3—34no—
72C2×C3×Dic6C2×D314yes—
73C2×C3×Dic6C2×D314yes—
74C2×#626—64no—
75C2×#624—64no—
76#3799—68no—
77#3799—68no—
78C32×Q16—62no—
79C2×C32×C8—24no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D311yescharacteristic
82#3755—62no—
83#3752—61no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—34no—
86C4⋊(C3⋊Dic3)—64no—
87#3750—61no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3×C3⋊C8—34no—
90C3×Dic12—28no—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3717—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)—22no—
96C2×C3⋊(C3⋊D3)—41no—
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3⋊(C3⋊Dic3)—41no—
104C3×C3⋊Dic3—21no—
105C32×Dic3—22no—
106C32×Dic3—41no—
107C2×#239—94no—
108C2×#235—91no—
109C2×C3×Q16—34no—
110#946—94no—
111C2×#215—94no—
112#924—91no—
113C2×Dic12C3⋊D311yescharacteristic, complemented
114C2×#204—91no—
115C3×C22⋊C8—91no—
116#835—94no—
117#821—94no—
118#821—91no—
119#821—184no—
120(C2×D3)⋊C8—34no—
121#816—91no—
122C22×C3⋊D3—61no—
123C22×C3⋊D3—64no—
124C22×C3⋊D3—124no—
125C22×C3⋊D3D1211yescharacteristic
126C22×C3×D3—34no—
127C32×Q8—32no—
128C32×Q8—61no—
129C32×Q8—61no—
130C32×Q8—61no—
131C2×C32×C4—24no—
132C2×C32×C4—61no—
133C2×C32×C4—61no—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—124no—
139C2×C3⋊Dic3D1211yescharacteristic
140C4⋊(C3⋊D3)—32no—
141C4⋊(C3⋊D3)—38no—
142C4⋊(C3⋊D3)—68no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)—64no—
146C4⋊(C3⋊D3)—124no—
147C4×C3⋊D3—21no—
148C4×C3⋊D3—21no—
149C4×C3⋊D3—64no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—34no—
153C3×C4×D3—64no—
154C3×C4×D3—64no—
155C3×Dic6—24no—
156C3×Dic6—24no—
157C3×Dic6—24no—
158C3×Dic6#6718yes—
159#626—616no—
160#624—616no—
161C32×C8—21no—
162C32×C8—24no—
163C32×C8—44no—
164C3⋊(C3⋊C8)—21no—
165C3×C3⋊C8—64no—
166#452—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C3⋊(C3⋊D3)—42no—
171C3×C3⋊D3—41no—
172C2×C3×Q8—34no—
173C2×C3×Q8—34no—
174C22×C3×C4—91no—
175C2×#67—181no—
176#239—188no—
177#239—188no—
178#235—182no—
179#235—182no—
180C2×D12—94no—
181C2×D12—91no—
182C2×D12—184no—
183C2×D12—184no—
184C2×C4×D3—34no—
185C2×C4×D3—181no—
186C2×C4×D3—184no—
187C2×Dic6C2×C3⋊D311yescharacteristic
188C2×Dic6C2×C3⋊D311yescharacteristic
189C3×Q16—68no—
190C2×C3×C8—24no—
191C2×C3×C8—91no—
192C2×C3×C8D3214yes—
193C2×C3×C8C2×C3⋊D311yescharacteristic
194#217—181no—
195#215—188no—
196(C2×D3)⋊C4—184no—
197C4⋊Dic3—94no—
198C4⋊Dic3—91no—
199C4⋊Dic3—184no—
200Dic3⋊C4—181no—
201Dic3⋊C4—184no—
202C2×C3⋊C8—34no—
203Dic12—22no—
204#204—182no—
205C22×C32—24no—
206C22×C32C2×D1211yescharacteristic
207C22×C32D3×D414yes—
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—62no—
212C2×C3⋊D3—68no—
213C2×C3⋊D3—128no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—121no—
216C2×C3⋊D3—124no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—244no—
219C2×C3⋊D3—244no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C32×C4—24no—
224C32×C4—24no—
225C32×C4—62no—
226C32×C4—121no—
227C32×C4D3×D414yes—
228C32×C4D3×D411yescharacteristic
229C32×C4D3×D411yescharacteristic
230C32×C4C2×#6714yes—
231C3⋊Dic3—21no—
232C3⋊Dic3—121no—
233C3⋊Dic3—124no—
234C3⋊Dic3—244no—
235C3×Dic3—28no—
236C3×Dic3—44no—
237C3×Dic3—64no—
238C2×#41—271no—
239C2×Q16—31no—
240C2×SD16—271no—
241C22⋊Q8—271no—
242#103—271no—
243#102—271no—
244C22⋊C8—91no—
245C33#45211yescharacteristic, complemented, Sylow
246C23×C3—91no—
247C22×D3—34no—
248C22×D3—184no—
249C22×D3—181no—
250C22×D3—364no—
251C3×Q8—38no—
252C3×Q8—64no—
253C3×Q8—64no—
254C3×Q8—64no—
255C2×C3×C4—24no—
256C2×C3×C4—64no—
257C2×C3×C4—64no—
258C2×C3×C4—91no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4C2×D3214yes—
262C2×C3×C4C22×C3⋊D311yescharacteristic
263#67—184no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—34no—
267C2×Dic3—184no—
268C2×Dic3—181no—
269C2×Dic3—364no—
270D12—98no—
271D12—92no—
272D12—188no—
273D12—1816no—
274D12—184no—
275D12—181no—
276D12—364no—
277C4×D3—64no—
278C4×D3—64no—
279C4×D3—184no—
280C4×D3—1816no—
281Dic6—21no—
282Dic6—21no—
283Dic6—21no—
284Dic6#63812yes—
285C3×C8—24no—
286C3×C8—21no—
287C3×C8—44no—
288C3×C8—181no—
289C3⋊C8—64no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32#88211yescharacteristic
294C2×C32#90411yescharacteristic
295C2×C32#90911yescharacteristic
296C2×C32#91814yes—
297C2×C32#91914yes—
298C2×C32#93714yes—
299C3⋊D3—41no—
300C3⋊D3—122no—
301C3⋊D3—128no—
302C3⋊D3—248no—
303C3×D3—124no—
304#41—542no—
305#41—542no—
306C2×Q8—31no—
307C2×Q8—31no—
308C2×D4—271no—
309C2×D4—541no—
310C22×C4—91no—
311C22×C4—541no—
312Q16—62no—
313SD16—542no—
314C2×C8—91no—
315C2×C8D3×C3⋊D311yescharacteristic
316C4⋊C4—271no—
317C4⋊C4—541no—
318C22⋊C4—541no—
319C22×C3—24no—
320C22×C3—181no—
321C22×C3—181no—
322C22×C3—181no—
323C22×C3D3×D1214yes—
324C22×C3D4×C3⋊D311yescharacteristic
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—64no—
328C2×D3—182no—
329C2×D3—188no—
330C2×D3—368no—
331C2×D3—364no—
332C2×D3—364no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—724no—
336C2×D3—724no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—68no—
340C3×C4—124no—
341C3×C4—181no—
342C3×C4#380214yes—
343C3×C4D3×#6714yes—
344C3×C4D4×C3⋊D311yescharacteristic
345C3×C4C2×#63811yescharacteristic
346Dic3—22no—
347Dic3—41no—
348Dic3—64no—
349Dic3—364no—
350Dic3—361no—
351Dic3—724no—
352C32—24no—
353C32#473311yescharacteristic, complemented
354C32#518714yescomplemented
355C23—91no—
356C23—541no—
357Q8—32no—
358Q8—61no—
359Q8—61no—
360Q8—61no—
361D4—272no—
362D4—544no—
363D4—541no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—544no—
370C2×C4—541no—
371C2×C4C2×D3×C3⋊D311yescharacteristic
372C8—21no—
373C8—181no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—361no—
378C2×C3#6380114yes—
379C2×C3#6380814yes—
380C2×C3#6392214yes—
381C2×C3#6412711yescharacteristic
382C2×C3#6412811yescharacteristic
383C2×C3#6414611yescharacteristic
384D3—124no—
385D3—362no—
386D3—368no—
387D3—728no—
388C22—181no—
389C22—181no—
390C22—181no—
391C22—542no—
392C22—1081no—
393C22—1081no—
394C22C3⋊D3×D1211yescharacteristic
395C4—62no—
396C4—121no—
397C4—181no—
398C4—1081no—
399C4#8902611yescharacteristic
400C4D3×#63811yescharacteristic
401C3—24no—
402C3#1059096614yescomplemented
403C3#1059288611yescharacteristic, complemented
404C2—361no—
405C2—1082no—
406C2#1172332911yescharacteristic
407C2#1172334111yescharacteristic
408C2#1172354011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89018C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C32×Dic12C2semidirect
3. #11723435C2semidirect
4. C2×#88795C2semidirect
5. #11722191C2semidirect
6. C3×#63647C2semidirect
7. #11722038C2semidirect
30. C2×C3×Dic12D3semidirect
37. #63647D3semidirect
113. C2×Dic12C3⋊D3semidirect
245. C33#452semidirect
353. C32#4733semidirect
354. C32#5187semidirect
402. C3#10590966semidirect
403. C3#10592886semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#890182
C2×C32×Dic122
#117234352
C2×#887952
#117221912
C3×#636472
#117220382
#105928863
#105909663